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4   BANACH ALGEBRAS WITH INVOLUTION. STAR ALGEBRAS       337
for all , where (ak) is a Hilbert basis of E. For each integer N we have there-

N

fore  \\un(av)\\2 ^ /? for all n; hence, letting n tend to + oo, we have

fc=0

lKa,)||2:g/J

fc=0

for all N; consequently (5.3.1), i;e J^2(E) and \\v\\l  p. Finally, for each
s > 0, there exists n0 such that \\um - un\\2 g e for all m^n0 and all n^nQy
and the same argument shows that \\v  un\\2^e for all n^n0. Hence
v = lim un in J?2(E).

n-* oo

It is easily verified that the Banach algebra &2(E)  not a star algebra
(Problem 13).

Remark

(15.4.8.6) Let (an) be a Hilbert basis of E. To each continuous operator
u e JSf (E) we may associate the double sequence (cmil) = ((w(an) | bmj) (some-
times called the infinite matrix of u with respect to (#)). We have seen
above that, if u is a Hilbert-Schmidt operator, then ||w||2 = ^ \cmn\2- Con-

m, n

versely, let (cmn) be any double sequence of complex numbers such that the
double family (|cmn|2) is summable. Then, for all , the vector bn = J] cmnam

m

is defined, and we have \\bn\\2 = ^ \cmn\2 (6.5.2). Furthermore, for each vector
x = Z tnan (with ||*||2 = X |^|2\ the vector u(x) = J tnbn is defined, and

n                                                  n                                                                n

||w(x)||2 ^(X Icmnl2>| ' II*II2; fr Ae Cauchy-Schwarz inequality shows that

\m, n           )

if h g A: we have

n-h

from which follows the absolute convergence of the series  n bn and the

n

inequality above for ||M(X)||. We have therefore defined a continuous endo-
morphism u of E (5.5.1), and it is immediately verified that

ZM*B)ll2 = I>mj2,

so that M e JSf 2(E).at lim \\u  un\\2 = 0. Hence, in view of